Answer to Question #23263 in Abstract Algebra for Melvin Henriksen

Question

Let R = End(Vk) where V is a vector space over a division ring k. Show that all ideals of R are linearly ordered by inclusion and idempotent.

Answer 1

If dimk V <∞, R isa simple ring.
Therefore, it suffices to treat thecase when V is infinite-dimensional.
The ideals of R are linearlyordered by inclusion.
To show that they are allidempotent, consider any ideal nonzero I.
There exists an infinite cardinal β <dimk V such that
I = {f ∈ R : dimk f(V ) < β}.
For any f ∈ I, let f' ∈ R be such that f' is theidentity on the f(V ), and zero on a direct complement of f(V). Clearly, f' ∈ I, and f= f'f. Therefore, f ∈ I^2, and wehave proved that I = I^2.